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# Aren’t Math Mistakes Beautiful?

It’s official. Here’s the research: having students analyze how math mistakes were made is more powerful than having them solve equation after equation.

I often show math professional learning communities a great film that Lucy West provided to me. Picture an 8^{th} grade math class in an urban school. One girl explains the equation she developed to describe the number of tiles that surround the edge of a swimming pool. Her equation is correct, but instead of saying, “Right!” the teacher says, “Thank you for sharing. Who has a different answer?”

Another boy comes to the front, places his equation and diagram on the overhead projector and says, **“After hearing her answer, I think I’m wrong, but I can’t figure out my mistake. Who can help me?” **And the other students, not the teacher, analyze the two threads of thinking and determine not just which is correct, but how the second student got off track. The teacher intervenes only to help the students make their diagrams clearer and ask for reasoning.

Powerful, isn’t it. A class where mistakes are

- Readily shared
- Seen as learning opportunities
- Analyzed by students

You know it’s higher-level thinking, the kind of instruction that meets the Common Core State Standards for Mathematical Practice. You know students need a deep understanding of concepts, not just procedures, to determine where mistakes come from. And now you’ve got the research to share with colleagues to further this great strategy. **Try it! You might:**

- Post three possible answers to a problem (just like those “detractor items” on standardized tests) and have students figure out what went wrong.
- Assign a “mistake” problem as homework on a specific concept you’re teaching students. For example, if they’re working on order of operations, try giving five “mistake” problems rather than a sheet of practice problems. What do students say about engagement? What do they learn?

Let me know!

I’ve been substitute teaching since this past November and I try to do something like this. Often, the reason is as simple as forgetting that one number is negative, not carrying the one, or multiplying instead of dividing. Other times, the explanation highlights flaws in the question (especially word problems). Students, particularly elementary and middle school ages, tend to think so far outside the box that there might as well not be a box. I’ve frequently ended up explaining that “Your answer isn’t _wrong_. However, it’s not the answer your teacher (or a test) is expecting. It’s not the sought-for answer for this particular situation.”

I hadn’t thought about its power for substitute teaching. Students who find mistakes are less likely to make similar ones later.

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